Theorem gagrp 18640

Description: The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
gagrp	⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Proof of Theorem gagrp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2736	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		eqid 2736	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 18639	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
5	4	simplbi 501	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
6	5	simpld 498	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  Vcvv 3398   × cxp 5534  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  +_gcplusg 16749  0_gc0g 16898  Grpcgrp 18319   GrpAct cga 18637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-map 8488  df-ga 18638

This theorem is referenced by:  gafo  18644  gass  18649  galcan  18652  gacan  18653  gapm  18654  gaorber  18656  gastacl  18657  galactghm  18750  sylow2alem2  18961